MA Day 34- February 22, 2013 Review for test #2 Chapter 11: Differential Multivariable Calculus

Paraboloids

Ellipsoids

Cones

Planes

Cylinders

The Idea: Describe f(x,y,z) by finding the surfaces on which it takes constant values.

Example:

Section 11.2: Limits and Continuity

Types of functions we will study: 1. Polynomials: 2. Rational functions: 3. Compound functions:

Types of functions we will study: 1. Polynomials: Continuous everywhere 2. Rational functions: 3. Compound functions:

Types of functions we will study: 1. Polynomials: Continuous everywhere 2. Rational functions: Continuous where defined 3. Compound functions:

Summary: Section 11.2 In future work you will be required to be able to determine whether or not a function is continuous at a point.

Section 11.3: Partial Derivatives

Geometrical interpretation of Partial Derivatives

There is a similar interpretation of partial derivatives.

Section 11.4: Tangent planes and linear approximations or On the differentiability of multivariable functions

The generalization of tangent line to a curve

Is tangent plane to a surface

Theorem: When f(x,y) has continuous partial derivatives at (a,b) then the equation for the tangent plane to the graph z = f(x,y) is

at P = (1,2,4)

(continuation of example)

Section 11.5 THE CHAIN RULE

Example:

Section 11.6 Directional Derivatives and the Gradient Vector

We need a practical way to compute this!

Question: Can the right-hand-side be written as a DOT PRODUCT?

Tangent planes to level surfaces

Significance of the Gradient Vector

Section 11.7: OPTIMIZATION As an application of our work in chapter 11, we set up the theory of how to find the local maximum and minimum values of f(x,y)

Section 11.7: OPTIMIZATION As an application of our work in chapter 11, we set up the theory of how to find the local maximum and minimum values of f(x,y)

So to find the local maxima and minima of a differentiable f(x,y) do the following:

(Continuation of example)